Distinguish between weak stationarity and strong stationarity. Explain the methods of testing for stationarity in a univariate time series model.

Introduction

Stationarity is a fundamental concept in time series analysis. A stationary time series is one whose properties do not depend on the time at which the series is observed. In econometrics, stationarity ensures that the statistical inferences made about the model are valid. There are two main types of stationarity: weak stationarity and strong stationarity. Understanding the distinction between them is important for choosing the appropriate tests and models.

Weak vs. Strong Stationarity

1. Weak Stationarity (Second-Order Stationarity)

A time series is weakly stationary if its:

• Mean is constant over time
• Variance is constant over time
• Covariance between values at two time points depends only on the distance or lag between them, not the actual time

Example: AR(1) process: Y_t = ρY_t-1 + ε_t (where |ρ| < 1) is weakly stationary.

2. Strong Stationarity (Strict Stationarity)

A time series is strongly stationary if the joint distribution of any set of observations is identical, regardless of when the observations are taken.

Formally, for any set of time indices t₁, t₂, …, t_k, the joint distribution of (Y_t1, Y_t2, …, Y_tk) is the same as (Y_t1+h, Y_t2+h, …, Y_tk+h) for all h.

Note: Every strongly stationary process is also weakly stationary, but the converse is not true.

Importance of Stationarity in Time Series

• Stationary time series are easier to model and forecast.
• Statistical properties (mean, variance) are stable over time.
• Most econometric models (like ARIMA) assume stationarity.

Testing for Stationarity in Univariate Time Series

There are several tests used to check stationarity in time series models. These tests help in identifying whether the series has a unit root or not (a sign of non-stationarity).

1. Augmented Dickey-Fuller (ADF) Test

The ADF test is widely used for testing unit roots. It involves estimating the following regression:

ΔY_t = α + βt + γY_t-1 + δ₁ΔY_t-1 + … + δ_pΔY_t-p + ε_t

Where:

• Δ is the difference operator
• γ is the coefficient of interest

Null Hypothesis (H₀): γ = 0 (non-stationary, unit root present)

Alternative Hypothesis (H₁): γ < 0 (stationary)

2. Phillips-Perron (PP) Test

Similar to the ADF test, but it makes a non-parametric correction to the test statistic to account for serial correlation and heteroscedasticity in the errors.

3. Kwiatkowski-Phillips-Schmidt-Shin (KPSS) Test

Unlike ADF and PP, KPSS has a different hypothesis structure:

Null Hypothesis (H₀): The series is stationary

Alternative Hypothesis (H₁): The series is non-stationary

Used often in conjunction with ADF for confirmation.

4. Visual Inspection

• Plotting the series over time helps in identifying trends or structural breaks.
• If the mean and variance appear constant, the series may be weakly stationary.

5. Autocorrelation Function (ACF) Plots

If the ACF of the series declines slowly, it indicates non-stationarity. A rapidly decaying ACF suggests stationarity.

Conclusion

Stationarity is essential for the correct specification and estimation of time series models. Weak stationarity assumes constant mean and variance, while strong stationarity assumes consistent joint distributions. Various tests like ADF, PP, and KPSS help in testing for stationarity. Understanding the type of stationarity and applying the right test ensures reliable forecasting and policy analysis using time series data.